Intensity of radiation reflected by a vertically inhomogeneous medium through multiple scattering

To facilitate the computation of the radiative intensity reflected, upon multiple scattering, by a vertically inhomogeneous medium, an implicit formula for integrating the invariant imbedding equation for Fourier-decomposed reflection function is derived starting with the formal solution: the height variations of the single-scattering albedo and the phase function characterizing the degree of inhomogeneity are thereby approximated by piece-wise, but continuous, linear functions of optical height , while the reflection function is approximated by a piece-wise quadratic polynomial in over each integration step. Using these approximations, the integration involved in the formal solution is then carried out analytically, yielding a correctortype formula for finding the reflection function at each step of . It is expected that this formula is capable of handling general cases of inhomogeneous media where both single-scattering albedo and phase function are allowed to vary continuously with height.Similar, but explicit expressions are also derived for the single and the second-order scattering solutions, with which the higher-order Fourier terms of reflection function are to be approximated, thereby enabling us to avoid the iterative process.     

Radiation transfer in inhomogeneous dispersive media

Equations for the angular density of radiation, reflected and transmitted intensities associated with radiation scattered by inhomogeneous dispersive media are obtained. The Padé approximant technique is used to calculate these intensities in inhomogeneous and homogeneous media. The results for the [0/1] Padé approximant lead to numerical results that compared with the exact results.     

Angular radiation transfer in inhomogeneous dispersive media

The equation of radiative transfer for an inhomogenous dispersive finite medium subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the angular distribution of radiation. Numerical results for the [0/1] Padé approximant lead to numerical results that compared with the exact results.     

Order-of-scattering of partially polarized radiation in inhomogeneous,anisotropically scattering atmospheres

A complete set of transfer equations required for the order-of-scattering analysis of partially polarized radiation in inhomogeneous, anisotropically scattering atmospheres is provided. The equations have been derived for both a local study using the radiative transfer equation and its associated auxiliary equation for the source-matrix, and a global study in terms of the scattering and transmission matrices; they account for the polarity of the scattering medium. Their derivations for the finite order scattering and the finitely cumulative scattering, in particular, have yielded important new equations expressing the invariance principles and the integro-differential recurrences for the scattering and transmission matrices. These novel expressions contain as a special case Bellmanet al's (1972) equations for the simpler case of isotropic scattering of unpolarized light in homogeneous atmospheres.     

Order-of-scattering of partially polarized radiation in inhomogeneous,anisotropically scattering atmospheres

Reciprocity and symmetry relationships, representing local invariants for the scattering phase-matrix, are derived for twelve cases of particle assemblies studied by van de Hulst (1957) including situations of scattering in an arbitrary direction, in the near forward and near backward directions. These relations are used to generate corresponding relations representing global invariants for the scattering and transmission matrices of atmospheres consisting of such assemblies. The latter relations are obtained from the matrix integro-differential equations for scattering and transmission; they apply to single scattering, any finite order of scattering, and after an arbitrary cumulation of scattering orders (finite or infinite). Our results are summarized in Tables I and II for general inhomogeneous atmospheres and for particular inhomogeneous atmospheres that are symmetrical with respect to their central level. The latter case includes homogeneous atmospheres as a special case. The largest set of local relations obtained contains three independent relations (called universal, reversal, exchange) which can further be combined to yield four additional dependent relations. This circumstance happens in three out of the above twelve cases. In the remaining cases fewer relations (both independent and dependent) remain valid. Likewise, a maximal set of three independent global relations is obtained for general inhomogeneous atmospheres; they too can be linearly combined to yield seven other dependent relations. For the symmetrically inhomogeneous atmospheres, three independent and seven dependent additional relations are obtained. On the basis of these tables, it becomes a trivial matter to provide the local and global invariants (both the independent and the dependent relations) for any assembly of particles and atmospheric inhomogeneity. A mixture of Rayleigh-Cabannes scattering by anisotropic molecules or extremely small particles and Mie scattering by large isotropic particles is considered for illustration. Lastly, the group properties of these invariants are studied.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the U.S. National Aeronautics and Space Administration.     

Moment method for the inverse radiative transfer in inhomogeneous media

In the present paper, an inverse problem of radiative transfer in an inhomogeneous plane medium of scattering albedo =₀ exp(-/s) is solved by a moment method. The results are compared with those obtained by Dunn (1983) using Monte Carlo method.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

The scattering map in the planar restricted three body problem

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the information already obtained for the homoclinic orbits. An example application to Astrodynamics is also proposed.     

On inhomogeneous scattering models of Titan's atmosphere

The spectrum of Titan from 4800 to 11 000 Å has many CH₄ absorption bands which cover a range of intensities of several orders of magnitude. Yet even the strongest of these bands in Titan's spectrum has considerable residual central intensity. Some investigators have concluded that these strong CH₄ bands must be highly saturated, but recent laboratory measurements of the bands made at room temperature show that curve-of-growth saturation is very small. At the presumed low pressures and temperatures in Titan's atmosphere, we show that saturation is very dependent on the band model parameters. However, in either a simple reflecting layer model or in a homogeneous scattering model saturation cannot be the principal cause of the filling in of these strong CH₄ bands if our best estimates of the band model parameters are correct. We find that an inhomogeneous scattering model atmosphere with fine “Axel dust” above most ot the CH₄ gas is needed to fill in the band centers. The calculated spectrum of one particular model of this class is compared to observations of Titan. Our essential conclusion is that Titan does have most of its scattering particles above most of the CH₄ gas which has an abundance of at least 2 km-am. This large abundance of CH₄ is necessary to produce the 6420-Å feature recently discovered in Titan's spectrum.     

Time-dependentX- andY-functions for anisotropically scattering inhomogeneous atmosphere by principle of invariance

The nonlinear integral equations forX- andY-functions have been developed for an inhomogeneous atmosphere scattering anisotropically using the principle of invariance. The anisotropy is represented by means of a phase function expressed in terms of finite-order Legendre polynomials.     

Non-coherent scattering in transfer problems in spherical shell media

The Combined Operations Method is utilised to solve Diffuse Reflection and Transmission Problems in inhomogeneous, isotropically and non-coherently scattering, spherical shell media. The source function is assumed to be frequency independent. TheN-solution of an auxiliary equation is sought, and tractable equations for the scattering and transmission functions are established. The solution of the problem for a scattering and emitting medium have been considered for a perfectly absorbing core.     

Optical properties of inhomogeneous media: T-matrix approach (Review)

In der vorliegenden Arbeit wird eine Übersicht darüber geben, wie man die effektive dielektrische Funktion, welche die optischen Eigenschaften heterogener Medien charakterisiert, berechnet. Für Inhomogenitäten kleiner als die Lichtwellenlange kann das entsprechende quasielektrostatische Problem in eine Integralgleichung umgeformt werden, welche die Anwendung des von der Festkörpertheorie her bekannten T-Matrix-Formalismus zur Berechnung von gestattet. Es werden unterschiedliche Näherungen für die T-Matrix diskutiert, die zu unterschiedlichen Effektiv-Medium-Theorien wie die Näherung der mittleren T-Matrix (ATA) oder die Näherung des kohärenten Potentials (CPA) führen.     

Radiative transfer for incoherent scattering in a semiinfinite inhomogeneous atmosphere containing energy sources

Byurakanskaya Astrophysical Laboratory. Translated from Astrofizika, Vol. 36, No. 4, pp. 563–574, October–December, 1993.     

Density dependence of the line source functions in scattering media

We have calculated source functions in a scattering medium in which the density changes according to the law ofN _e (r)～r ⁿ wheren takes the value from ?3 to +3(1) andN _e (r) is the electron density. We have assumed that the media consist of electrons and we have also considered a geometrically extended media in which the outer radii are 2, 3, 5 times the inner radius. The source functions obtained are completely due to electron scattering. It is found that the source function varies considerably for different variations of density changes fromn=?3 to +3. In the case of density variation withn=?3 and ?2, the source functions do not increase with optical depth considerably, but whenn=?1, 0, they rise slowly with the increase in optical depths and whenn=1 to 3 there is a steep rise in the source functions with the optical depth increasing towards the center of the star.     

Sobolev's Φ-function of radiative transfer in planar and spherical scattering media

In recent years Sobolev's -function of radiative transfer has been discussed in connection with the resolvent of Milne's integral equation such that it plays an important role in the determination of the radiation field in semi-infinite (or finite) atmospheres with internal sources (cf. Sobolev, 1963). In the present paper, the part of Sobolev's -function in plane-parallel and spherical, isotropically scattering, atmospheres with internal source distribution is investigated from analytical and numerical aspects. With the aid of invariant imbedding (cf. Bellmanet al., 1968), we computed Sobolev's -function of Milne's integral equation for the planar case by solving the Cauchy system for the auxiliary function and Chandrasekhar'sX- andY-functions. The corresponding -function for the spherical case is readily obtained from the for the planar case.Investigation supported by the National Science Foundation under Grant No. GP 29049, the Atomic Energy Commission, Division of Research under Contract No. AT(04-3)-113, Project 19, and the National Institutes of Health under Grant No. 16197-05.     

The international planetary patrol program: An assessment of the first three years

Day-to-day and hour-to-hour changes in the large-scale atmospheric and surface features of the planets can now be studied more effectively than previously possible. Since 1969 a network of observatories has obtained almost uninterrupted photographic coverage during all apparitions of Jupiter and Mars, plus some of Venus. Patrol films and catalogues are available to the scientific community. Recent or current analyses include the distribution and motion of clouds on Mars, the development and decay of Martian dust storms, the seasonal, diurnal and random fluctuations in contrast between adjacent light and dark regions on Mars, the detection of vertical shear in the Jovian atmosphere, the longitudinal oscillation of the Red Spot, the dependence of rotation period on xenographic latitude and on time, the eruption and spread of SEB disturbances, and the retrograde circulation of the Venus cloud deck.